On 30 Mai, 02:56, "Peter Webb" <webbfam...@DIESPAMDIEoptusnet.com.au>wrote:> >> Grant the existence of two natural numbers m and n such that m/n = sqrt> >> (2). Then falsify it.> >> Grant the existence of a largest natural. Then falsify it.> >> Grant the existence of a largest prime number. Then falsify it.> >> Grant the existence of all natural numbers. Then falsify it.>> >> All these proofs are proofs by contradiction.>> >> *************************>> >> How do you falsify the existence of the set of all Natural numbers in ZF> >> ?> >> ZF includes an axiom of infinity, which pretty much directly guarantees> >> that> >> there is an infinite set of all finite ordinals.>> > What about classical arithmetics with an axiom that sqrt(2) is a> > rational number?>> This does not answer my question.>> But I will answer yours for you. If you create a version of "classical" (=> "standard" ?) arithmetic with an axiom that sqrt(2) is rational, it would > be> inconsistent, and hence useless.

Same with the axiom of infinity: ?There exists a complete linearinfinite set? is a self-contradictory similar to ?there exists a pairof natural numbers, a and b, such that b^2 = 2a^2.>> Now how about answering my question. How do you falsify the existence of the> set of all Natural numbers in ZF, as you claimed you could?

It is simple: ... classical logic was abstracted from the mathematicsof finite sets and their subsets .... Forgetful of this limitedorigin, one afterwards mistook that logic for something above andprior to all mathematics, and finally applied it, withoutjustification, to the mathematics of infinite sets. [Hermann Weyl,"Mathematics and logic: A brief survey serving as a preface to areview of The Philosophy of Bertrand Russell", American MathematicalMonthly 53: 2?13]

Therefore: Either [*] holds or the set is not complete but allows forextension.Then we have onlyEn Am: m =< n ==> Am En m =< n [**]because not all elements are readily available.That is called a potentially infinite set. But in this case there isno chance to prove uncountability.This had already been recognized by the late Alexander Zenkin, one ofthe brave scientists who dared to condemn this hypocriticalbehaviour:Cantor's 'paradise' as well as all modern axiomatic set theory isbased on the (self-contradictory) concept of actual infinity. Cantoremphasized plainly and constantly that all transfinite objects of hisset theory are based on the actual infinity. Modern AST-people try topersuade us to believe that the AST does not use actual infinity. Itis an intentional and blatant lie, since if infinite sets, X and N,are potential, then the uncountability of the continuum becomesunprovable, but without the notorious uncountablity of continuum themodern AST as a whole transforms into a long twaddle about nothing.

Resume: The internal contradiction in set theory is veiled by mixingup potential and actual infinity. That is the reason why set theoristsusually refuse to specify which infinity they apply. Most even pretend(or profess) not to know the difference.